A new formulation of the nuclear

level density by spin and parity

Richard B. Firestone

University of California, Department of Nuclear Engineeering,

Berkeley, CA

IntroductionLevel densities are important for nuclear reaction calculations. They are typically

calculated with the Constant Temperature (CT) and Back Shifted Fermi Gas (BSFG)

formulations. Each formulation has two parameters that are compiled in the IAEA

Reaction Input Parameter Library (RIPL).

• Both models calculate total level densities independent of spin/parity.

• Spin distribution is calculated with a spin distribution function which is

dependent on the spin cutoff parameter σc.

• Parity is not calculated.

All reaction experiments populate a subset of spins and parities.

• The spin distribution function is invalid at low excitation energy.

• Level density parameters and σc are calculated using low energy level data.

• At low energies for most nuclei the levels are predominantly one parity.

• Comparing reaction data to RIPL total level densities is invalid.

Today I will discuss a modification of the CT formulation that is

• Valid for all spins and parities

• Applicable to nuclei with no resonance data

Level Density Formulations

Constant Temperature (CT)

� � = exp � − ��

� �, =�( ) ∙ �(�)

E – level energyN(E) – level sequence numberE0 – back shift parameterT – critical energy for breaking

nucleon pairsT. Ericson NP 11, 481 (1959)T. Ericson NP 11, 481 (1959)T. Ericson NP 11, 481 (1959)T. Ericson NP 11, 481 (1959)

Back Shifted Fermi Gas (BSFG)

� �, = �( )234 2 6(� − �7)

12 2896�.:;(� − �7)7.:;

a – level density parameterE1 – back shift parameterσc – spin cutoff parameter

T.D. Newton, Can. J. Phys. 34, 804 (1956),T.D. Newton, Can. J. Phys. 34, 804 (1956),T.D. Newton, Can. J. Phys. 34, 804 (1956),T.D. Newton, Can. J. Phys. 34, 804 (1956),A.G.W. Cameron, Can, J. Phys. 36, 1040(1958)A.G.W. Cameron, Can, J. Phys. 36, 1040(1958)A.G.W. Cameron, Can, J. Phys. 36, 1040(1958)A.G.W. Cameron, Can, J. Phys. 36, 1040(1958)

E0, T, a,E1, and σc are parameters, see T. von T. von T. von T. von EgidyEgidyEgidyEgidy and D. and D. and D. and D. Bucurescu, Phys. Rev. C 72,

044311 (2005) and RIPL-3.

Calculation of these parameters is

described by Gilbert and Cameron, Can. J.

Phys. 43, 1446 (1965).

Other models take into account shell

and collective effects or apply combinatorial

methods.

Spin Distribution Function

� =2 + 1

289: 234 −

( +12):

289:

T. Ericson, Adv. Phys. 9, 425 (1960)T. Ericson, Adv. Phys. 9, 425 (1960)T. Ericson, Adv. Phys. 9, 425 (1960)T. Ericson, Adv. Phys. 9, 425 (1960)

CT Formulation

� � = exp � − ��

� �, =�( ) ∙ �(�)

For the rest of this talk I will discuss the CT

formulation.

Basic assumptions

• The total level density increases exponentially with a temperature T and a

back shifted level energy (E−E0).

• The level density for each spin has the same T and E0.

• T is a fundamental parameter that can be accurately described, for example,

as the BCS critical temperature (Moretto et al, Journal of Physics:

Conference Series 580 (2015) 012048) for A>100.

GH =2∆�

3.53, ∆JK≈ 12M

N7

:, ≈ 6.80MN7

:

∆JK is the gap parameter from the liquid drop model (Bohr and Mottelson)

• E0 is a meaningless fitting parameter

• Spin is a separable function f(J).

• Parity is ignored.

• The useful level energy range of the CT formulation is not apparent.

How are the CT parameters derived?

The level number �(�OP)

at the neutron separation

energy is

� �OP=

Q0 ∙ � 1 2⁄ S

where D0 is the s-wave n-

capture level spacing and

� 1 2⁄ S � 0.5 ∙ ��1 2⁄ �From the spin distribution

function.

T and E0 are derived

solely from an exponential

fit to the low-lying levels.

T is independent of E0

Level sequence numbers increase exponentially for all

nuclei. This is a fundamental observation of RIPL.

First few

levels ignored

E0=intercept

Level spacing increases exponentially for all Jππππ

Selected spins can also

be fit to an exponential.

Bandhead energies

can be fit to ±108

keV, χ2/f=1.0

CT-JPI Formulation

• Define E0 as the Yrast energy for each spin and parity.

This requires N(E)=1 for the first occurrence of each Jπ value. Natural value

• T is a constant temperature fit to all spin and parity sequences

This is a primary tenet of the CT formulation

• T(Jπ)exp and D(Jπ)exp, the level spacing at Sn for Jπ, are determined by an

exponential fit of N(E) to E-E0(Jπ) for the resonance energies E of each Jπ.

• D(Jπ)calc is determined from the spin distribution function for Jπ when no

resonance data exists where the spin cutoff parameter σc is chosen so that

D(Jπ)calc coincides with experiment.

This formulation provides a self consistent level density calculation

requiring only one free parameter, T.

The CT-JPI formulation differs from the CT formulation in that E0 is

a physical parameter and T is determined from resonance data

rather than level data.

The CT-JPI Formulation – 57Fe

CT-JPI fit to experimental resonance data for 57Fe. The negative parity levels

have constant T, positive parity levels have variable T.

RIPL: T=1.31 MeV, E0=−1.41 MeV

1/2−−−− T=1.685(5) MeV

E0=0.0 MeV

3/2−−−− T=1.690(4) MeV

E0=0.0144 MeV

1/2++++ T=1.231(5) MeV

3/2++++ T=1.132 (4) MeV

5/2+ T=1.27(14) MeV

Experimental Resonance Data (292 s-, p-, d-wave Resonances)

E0=3.298 MeV

E0=2.836 MeV

E0=2.505 MeV

The CT-JPI Formulation – 57FeSpin Distribution Calculations

0

5

10

15

20

1.5 2.5 3.5 4.5

∆∆ ∆∆T

(%)

σσσσc

σσσσc=2.56

Spin distribution

ratioExperiment Calculated

� 1/2� 3/2U 1.56(1) 1.59

� 5/2� 3/2U 1.16(13) 1.02

T=1.664 MeV, All J, π=−T=1.131 MeV, J=1/2+,5/2+,9/2+,…

T=1.28 MeV, J=3/2+,7/2+,11/2+,…

σc was varied to get constant T

σc=3.17 (von Egidy)

Shaded values from experiment, others from spin

distribution function. I get three temperatures

averaging all values.

The CT-JPI Formulation – 57FeFor higher spins T and D(Jπ), the level spacing at Sn for levels of Jπ, calculated from

the spin distribution function diverge.

The simple form of the spin-dependent level density is expected to fail for large

values of the angular momentum. J.R. Huizenga and L.G. Moretto, Ann. Rev.

Nucl. Sci. 22, 427 (1972).

Jππππ T(calc)

MeV

D(Jππππ)(calc)

keV

T(syst)

MeV

D(Jππππ)(syst)

keV

13/2+ 1.73 143 1.13 24

15/2+ 1.61 373 1.28 200

17/2+ 2.97 1278 1.13 74

19/2+ 3.24 4908 1.28 970

21/2+ 10.7 22044 1.13 364

13/2- 1.91 164 1.66 74

15/2- 2.50 450 1.66 202

17/2- 3.70 1463 1.66 658

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1/2 5/2 9/2 13/2 17/2 21/2D

(Jππ ππ)

Spin

π=+π=+π=+π=+π=−π=−π=−π=−Calculated

The spin distribution function fails for

Jπ>11/2. A constant temperature

works better for high spin but also σc

could be varied.

Total level density– 57Fe

The total level density is determined by summing over all Jπ values. Agreement

with RIPL CT calculation and Oslo data are good. VS

VN⁄ = 1.2at 9 MeV.

The CT-JPI Formulation – 236U

0

5

10

15

20

1.5 2.5 3.5 4.5

∆∆ ∆∆T

(%)

σσσσc

σσσσc=2.40

D(3)/D(4)=1.51 (expt), =1.56(calc)

σc=4.74 (von Egidy)

932 Resonances

The CT-JPI Formulation – 236USpin distribution function fit Assumption: D(3−,4−)=D(3+,4+).

W=0.443(5) MeV

T= 0.44(1), von Egidy

T=0.393 (RIPL)

TBCS=0.443 MeV

Spin distribution function fails at J≥6

Note: The Yrast GS band gives T=0.516(14)

MeV. The Yrare band gives T=0.443(16) MeV

and is used in the CT-JPI fit.

236U Wigner fluctuations in TThe average level spacing is expected to follow a Wigner distribution

XY

Q=

Z

2

Y

Q234 −

Z

4

Y

Q

:

, [V� [S7V X YQ � 234 � Y

Q , [V \ [S7V

where Y � �[S7],V � �[],V , Q � 1 ���, , Z�⁄Small variations in T are due to Wigner

fluctuations in the positions of the Yrast levels.

Assuming T=0.443 MeV, the effective Yrast

energies can be calculated. Fluctuations in the

Yrast energies, Eyrast-Eeff, can be compared with

the Wigner distribution. Reasonable

agreement it obtained for 236U.

Fluctuations in level spacing give modest

agreement with Wigner distribution (missing

levels?).

The validity of the Wigner distribution near the

GS has been shown by Brody et al, Rev. Mod.

Phys. 54, 385 (1981).

Comparison of Wigner distribution

with experimental E0 fluctuations

and level separations.

E0 should be the effective value

calculated from a constant T.

The CT-JPI Formulation – 236U

Level densities, π−/π+=1.33 at 6.5 MeV.

Improved 236U Yrast Data

• 0−−−− level

No 0− levels are known in 236U. The only 0- levels known in the

actinides are at 1311.51 keV (236Pu) and 1410.75 keV (238Pu).

Assuming T=0.443(5) MeV, E(0−)=1330±140 keV.

• 1++++ level

The Yrast 1+ level from ENSDF at 1791.3 keV gives a low T=0.380 MeV.

Levels at 957.9- and 960.3-keV were assigned (2+) in ENSDF which is

improbable for a Wigner distribution. The 957.9-keV level is not

populated in average resonance capture from Jπ=3−,4−. This level is

consistent with an Yrast 1+ assignment and gives T=0.440 MeV.

• 5++++ Level

Lowest candidate state is at 1093.8 keV with Jπ=(2,5)+ in ENSDF. Level

is not feed by 236Pa β− decay (1−) suggesting the higher spin is likely

and gives T=0.438 MeV.

Conclusions• The CT-JPI formulation can be used to calculate complete level schemes up to

at least the neutron separation energy.

This is a fundamental property of all nuclear level schemes

• The E0 parameter restricts the Jπ level density to levels above the Yrast level.

The parity distribution comes naturally from this formulation

• The temperature T derived from resonance and low energy data are the same.

The CT-JPI model is valid at all excitation energies

• The spin cutoff parameter, σc, is directly determined.

• The spacing of all levels follows a Wigner distribution.

• Calculation of the level density for high spin states is uncertain.

• More nuclei need to be studied to determine the range of applicability of the

CT-JPI formulation.

Epiphany: For nuclei without resonance data the temperature can be derived

from the low levels and combined with the Yrast data to determine the level

density.

238U Level Density

Fit to levels #3-26

T=0.440 MeV

TBCS=0.441 MeV

238U Yrast E0 values and Jππππ level spacing at Sn

JE0(ππππ=+)

keV

D(J+)

eV

E0(ππππ=−−−−)

keV

D(J−−−−)

eV

D(J)-exp

eV

D(J)-calc

eV

0 927.21 3.01 - - 3.01 3.19

1 - - 680.11 1.71 1.71 2.17

2 966.13 3.29 950.12 3.17 1.61 1.49

3 1059.66 4.07 731.93 1.93 1.31 1.39

4 1056.38 4.04 1028 3.79 1.95 2.07

5 1232 6.02 862.64 2.37 1.71 1.62

The experimental level spacing

at Sn is determined from the CT-JPI

model parameters.

Agreement with the spin

distribution function is good for

σc=3.98. No 1+ or 0− levels are

predicted below Sn=6154.3 kev.

Comparison of 238U level spacing

with Wigner distribution.